Kahan p9 Example

Average Error: 20.3 → 5.7

Time: 6.5s

Precision: binary64

Cost: 3400

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

↓

\[\begin{array}{l} t_0 := x \cdot x + y \cdot y\\ t_1 := \frac{x \cdot x - y \cdot y}{t_0}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{t_1} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{t_0}\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* y y))) (t_1 (/ (- (* x x) (* y y)) t_0)))
   (if (<= y -2e+160)
     -1.0
     (if (<= y -1.8e-153)
       (* (/ 1.0 t_1) (* t_1 t_1))
       (if (<= y 1.18e-172) 1.0 (/ (* (- x y) (+ x y)) t_0))))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double t_1 = ((x * x) - (y * y)) / t_0;
	double tmp;
	if (y <= -2e+160) {
		tmp = -1.0;
	} else if (y <= -1.8e-153) {
		tmp = (1.0 / t_1) * (t_1 * t_1);
	} else if (y <= 1.18e-172) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) * (x + y)) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * x) + (y * y)
    t_1 = ((x * x) - (y * y)) / t_0
    if (y <= (-2d+160)) then
        tmp = -1.0d0
    else if (y <= (-1.8d-153)) then
        tmp = (1.0d0 / t_1) * (t_1 * t_1)
    else if (y <= 1.18d-172) then
        tmp = 1.0d0
    else
        tmp = ((x - y) * (x + y)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

public static double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double t_1 = ((x * x) - (y * y)) / t_0;
	double tmp;
	if (y <= -2e+160) {
		tmp = -1.0;
	} else if (y <= -1.8e-153) {
		tmp = (1.0 / t_1) * (t_1 * t_1);
	} else if (y <= 1.18e-172) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) * (x + y)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

↓

def code(x, y):
	t_0 = (x * x) + (y * y)
	t_1 = ((x * x) - (y * y)) / t_0
	tmp = 0
	if y <= -2e+160:
		tmp = -1.0
	elif y <= -1.8e-153:
		tmp = (1.0 / t_1) * (t_1 * t_1)
	elif y <= 1.18e-172:
		tmp = 1.0
	else:
		tmp = ((x - y) * (x + y)) / t_0
	return tmp

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

↓

function code(x, y)
	t_0 = Float64(Float64(x * x) + Float64(y * y))
	t_1 = Float64(Float64(Float64(x * x) - Float64(y * y)) / t_0)
	tmp = 0.0
	if (y <= -2e+160)
		tmp = -1.0;
	elseif (y <= -1.8e-153)
		tmp = Float64(Float64(1.0 / t_1) * Float64(t_1 * t_1));
	elseif (y <= 1.18e-172)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / t_0);
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

↓

function tmp_2 = code(x, y)
	t_0 = (x * x) + (y * y);
	t_1 = ((x * x) - (y * y)) / t_0;
	tmp = 0.0;
	if (y <= -2e+160)
		tmp = -1.0;
	elseif (y <= -1.8e-153)
		tmp = (1.0 / t_1) * (t_1 * t_1);
	elseif (y <= 1.18e-172)
		tmp = 1.0;
	else
		tmp = ((x - y) * (x + y)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -2e+160], -1.0, If[LessEqual[y, -1.8e-153], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.18e-172], 1.0, N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Target

Original	20.3
Target	0.1
Herbie	5.7

\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if y < -2.00000000000000001e160

   1. Initial program 64.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]
      rational.json-simplify-1 [=>] 64.0	\[ \frac{\color{blue}{\left(y + x\right)} \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-61 [=>] 64.0	\[ \frac{\color{blue}{x \cdot x - y \cdot y}}{x \cdot x + y \cdot y} \]

   3. Taylor expanded in x around 0 0

      \[\leadsto \color{blue}{-1} \]

   if -2.00000000000000001e160 < y < -1.7999999999999999e-153

   1. Initial program 1.5

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   2. Simplified 1.5

      \[\leadsto \color{blue}{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}} \]
      Proof

      [Start] 1.5	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-2 [=>] 1.5	\[ \frac{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]
      rational.json-simplify-1 [=>] 1.5	\[ \frac{\color{blue}{\left(y + x\right)} \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-61 [=>] 1.5	\[ \frac{\color{blue}{x \cdot x - y \cdot y}}{x \cdot x + y \cdot y} \]

   3. Applied egg-rr 1.5

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}} \cdot \left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y} \cdot \frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)} \]

   if -1.7999999999999999e-153 < y < 1.17999999999999999e-172

   1. Initial program 28.8

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   2. Simplified 28.8

      \[\leadsto \color{blue}{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}} \]
      Proof

      [Start] 28.8	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-2 [=>] 28.8	\[ \frac{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}{x \cdot x + y \cdot y} \]
      rational.json-simplify-1 [=>] 28.8	\[ \frac{\color{blue}{\left(y + x\right)} \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-61 [=>] 28.8	\[ \frac{\color{blue}{x \cdot x - y \cdot y}}{x \cdot x + y \cdot y} \]

   3. Taylor expanded in x around inf 15.6

      \[\leadsto \color{blue}{1} \]

   if 1.17999999999999999e-172 < y

   1. Initial program 1.5

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}} \cdot \left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y} \cdot \frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	1356

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	11.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-157}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	21.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))